{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:20Z","timestamp":1753893800226,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>A triple $(A,B,C)$ of dice is called nontransitive if each of $P(A&lt;B)$, $P(B&lt;C)$, and $P(C&lt;A)$ is greater than $\\frac12$ and called balanced if $P(A&lt;B)=P(B&lt;C)=P(C&lt;A)$. From the result of Trybu\u0142a, it is known that $P(A&lt;B)$ is less than $\\frac{-1+\\sqrt{5}}{2}$, the golden ratio, for every balanced nontransitive triple $(A,B,C)$ of dice. Schaefer asked whether this upper bound is tight, and Hur and Kim conjectured that the upper bound can be reduced to $\\frac12+\\frac19$. In this paper, we characterize all possible probabilities $P(A&lt;B)$ for balanced nontransitive triples $(A,B,C)$ of dice. Precisely, we prove that, for every rational $\\frac12 &lt;q&lt;\\frac{-1+\\sqrt{5}}{2}$, there exists a balanced nontransitive triple $(A,B,C)$ of dice with $P(A&lt;B)=q$, which disproves Hur and Kim's conjecture and answers Schaefer's question.\r\nWe also characterize all triples $(m,n,\\ell)$ of positive integers such that there exists a balanced nontransitive triple $(A,B,C)$ of dice, where $A$, $B$, and $C$ are $m$-sided, $n$-sided, and $\\ell$-sided dice, respectively. This generalizes Schaefer and Schweig's result showing the existence of a balanced nontransitive triple of $n$-sided dice for every $n\\ge 3$.<\/jats:p>","DOI":"10.37236\/11918","type":"journal-article","created":{"date-parts":[[2024,1,26]],"date-time":"2024-01-26T09:52:05Z","timestamp":1706262725000},"source":"Crossref","is-referenced-by-count":0,"title":["Balanced Nontransitive Dice: Existence and Probability"],"prefix":"10.37236","volume":"31","author":[{"given":"Dohyeon","family":"Kim","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ringi","family":"Kim","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wonjun","family":"Lee","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yuhyeon","family":"Lim","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yoojin","family":"So","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2024,1,26]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v31i1p21\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v31i1p21\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,1,26]],"date-time":"2024-01-26T09:52:06Z","timestamp":1706262726000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v31i1p21"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,1,26]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2024,1,12]]}},"URL":"https:\/\/doi.org\/10.37236\/11918","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2024,1,26]]},"article-number":"P1.21"}}